## Things that can and things that can't be done in PRA (1998)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Kohlenbach98thingsthat,

author = {Ulrich Kohlenbach},

title = {Things that can and things that can't be done in PRA},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper

### Citations

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Citation Context ...ions as fn := λm.f(n,m). However the use of variables f 0(0)...(0) is more convenient since it avoids the use of the λ-operator in many cases. n 3functionals of type level ≤ 2 in the sense of Kleene =-=[7]-=- (i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. [6](II.1) or [22]; we do not include higher type primitive recursion in the sense of [5]). We also have a sch... |

220 |
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Citation Context ...nalysis) can be carried out in systems T which are conservative Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 over primitive recursive arithmetic PRA (see [=-=25-=-] for a systematic account). This is of interest for mainly two reasons 1) If a 0 2 -sentence A is provable in T and the conservation of T over PRA has been established proof-theoretically, then one ... |

162 |
Uber eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
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Citation Context ...ot the iteration functional (It) it (0; y; f) = y; it (x + 1; y; f) = f(x; it (x; y; f)); although it is primitive recursive in the sense of Kleene (and not only in the extended sense of Godel [5], `=' is equality between natural numbers). We call the resulting system PRA 2 . On easily shows that PRA 2 is a denitorial extension of PRA 2 + (It). 2 So we could have used also variables and quant... |

70 |
Proof-theory: some applications of cut-elimination
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Citation Context ...g equations for all primitive recursive functionals of type level 2 in the sense of Kleene [7] (i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. [6](II.1) or [=-=-=-21]). We also have a schema of quantier-free induction (w.r.t. to this extended language) and -abstraction for number variables, i.e. (y:t[y])x = t[x]; x; y tuples of the same length. So PRA 2 is the ... |

57 | Analysing proofs in analysis
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Citation Context ... make the schema equivalent to the single second-order sentence () 8(a n ) [0; 1] d BW(a n ): 1 For precise formalizations of these principles in systems based on number and function variables see [1=-=2]-=- on which the present paper partially relies. We slightly deviate from the notation used in [12] by writing (PCM),(PCMar) instead of (PCM2),(PCM1). 2 It is well-known by the work on program of reverse... |

47 |
Theories of finite type related to mathematical practice
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- 1977
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Citation Context ...and λ-abstraction for number variables, i.e. (λy.t[y])x = t[x], x,y tuples of the same length. So PRA 2 essentially is the second-order fragment of the (restricted) finite type system PA ω | \ from =-=[3]-=-. It is clear that the resulting system PRA 2 is conservative over PRA. We often write 1 instead of 0(0). Another option is to impose a restriction on the type-2-functionals which are allowed. We incl... |

44 |
Introduction to Metamathematics, North-Holland
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Citation Context ...ons). 2 We have the axioms and rules of many-sorted classical predicate logic as well as symbols and dening equations for all primitive recursive functionals of type level 2 in the sense of Kleene [7=-=-=-] (i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. [6](II.1) or [21]). We also have a schema of quantier-free induction (w.r.t. to this extended language) and ... |

43 |
On a number theoretic choice schema and its relation to induction
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- 1970
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Citation Context ...s 0 1 -CA and therefore every function parameter-free instance of the principle of 0 1 -collection principles 0 1 -CP. Hence PRA+ 0 1 -CP is a subsystem of PRA 2 + AC 0;0 -qf+PCM . However from [17] we know that there exists an instance of 0 1 -CP which cannot be proved 9 In the proof of theorem 4.27 from [9], AC 0;1 -qf is only needed to derive the strong sequential version WKLseq of WKL. 10 ... |

41 | Partial realizations of Hilbert’s program
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Citation Context ...er PRA has been establishedsnitistically (which is possible for mathematically strong systems T (see [22],[8]), then all the mathematics which can be carried out in T has asnitistic justication (see [=-=24]-=-,[25] for a discussion of this). In this paper we exhibit a sharp boundary betweensnistically reducible parts of analysis and extensions which provably go beyond the strength of PRA. More precisely we... |

40 |
Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
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(Show Context)
Citation Context ...ch are on the one handmathematically very strong but on the other hand are still conservative over PRA (and even much weaker systems) have been developed by the author in a series of papers (see e.g. =-=[8]-=-,[9] and – for a general survey – [15]). These facts are of interest for mainly two reasons 1) If a Π0 2-sentence A is provable in T and the conservation of T over PRA has been established proof-theor... |

35 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals, Archive for Mathematical Logic 36
- Kohlenbach
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Citation Context ...quite restricted complexity or rate of growth (compared to merely being primitive recursive). In fact in a series of papers we have shown that in many cases even a polynomial bound is guaranteed (see =-=[9]-=-,[11],[14] among others). 2) One can argue that PRA formalizes what has been calledsnitistic reasoning (see e.g. [26]). If the conservation of T over PRA has been establishedsnitistically (which is po... |

31 | Hierarchies of provably recursive functions - Fairtlough, Wainer - 1998 |

30 |
On n-quantifier Induction
- Parsons
- 1972
(Show Context)
Citation Context ...s PRA+ 0 2 -IA. Denition 3.6 By T n we denote the fragment of Godel's calculus T of primitive recursive functionals in allsnite types where one only has recursor constants R with deg() n (see [19] for details). Corollary 3.7 Let A 8x9yA 0 (x; y) be a 0 2 -sentence in L(PRA). Then the following rule holds: 8 > > > > > > > > : PRA 2 +AC 0;0 -qf+WKL+PCM +BW +A-A +Limsup ` 8x9yA 0 (x; y) ) one... |

28 |
Fragments of arithmetic
- Sieg
- 1985
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Citation Context ...PRA formalizes what has been calledsnitistic reasoning (see e.g. [26]). If the conservation of T over PRA has been establishedsnitistically (which is possible for mathematically strong systems T (see =-=[22-=-],[8]), then all the mathematics which can be carried out in T has asnitistic justication (see [24],[25] for a discussion of this). In this paper we exhibit a sharp boundary betweensnistically reducib... |

17 | Proof theory and computational analysis
- Kohlenbach
- 1998
(Show Context)
Citation Context ...tricted complexity or rate of growth (compared to merely being primitive recursive). In fact in a series of papers we have shown that in many cases even a polynomial bound is guaranteed (see [9],[11],=-=[14]-=- among others). 2) One can argue that PRA formalizes what has been calledsnitistic reasoning (see e.g. [26]). If the conservation of T over PRA has been establishedsnitistically (which is possible for... |

17 |
Subrecursion: Functions and Hierarchies, Oxford Logic Guides
- Rose
- 1984
(Show Context)
Citation Context ...s a bounded search operator and bounded recursion { uniformly in function parameters { on the ground type (see [9]). 3 EA 2 is the restriction of PRA 2 to elementary recursive function(al)s only (see =-=[20-=-] for a denition of `elementary recursive functional'). Remark 1.1 In contrast to the class of primitive recursive functions, there exists no Grzegorzcyk hierarchy for primitive recursive functionals ... |

14 |
Reverse mathematics
- Simpson
- 1985
(Show Context)
Citation Context ...ar yields that relatively to EA 2 the principle PCM: 8f PCM(f) implies CA ar . For RCA 0 instead of EA 2 this implication is stated in [4]. A proof (which is dierent from our proof) can be found in [2=-=3-=-]. Proposition 4.2 and proposition 5.5 together yield (using the fact thatsnitely many instances of 0 1 - d AC can be coded into a single function and number parameter-free instance) Theorem 5.8 Let ... |

13 | Foundational and mathematical uses of higher types
- Kohlenbach
(Show Context)
Citation Context ...very strong but on the other hand are still conservative over PRA (and even much weaker systems) have been developed by the author in a series of papers (see e.g. [8],[9] and { for a general survey { =-=[15-=-]). These facts are of interest for mainly two reasons 1) If a 0 2 -sentence A is provable in T and the conservation of T over PRA has been established proof-theoretically, then one can extract a pri... |

12 |
E#ective bounds from ine#ective proofs in analysis: an application of functional interpretation and majorization
- Kohlenbach
- 1992
(Show Context)
Citation Context ...ormalizes what has been calledsnitistic reasoning (see e.g. [26]). If the conservation of T over PRA has been establishedsnitistically (which is possible for mathematically strong systems T (see [22],=-=[8-=-]), then all the mathematics which can be carried out in T has asnitistic justication (see [24],[25] for a discussion of this). In this paper we exhibit a sharp boundary betweensnistically reducible p... |

12 | Elimination of Skolem functions for monotone formulas in analysis - Kohlenbach - 1998 |

10 |
Note on the fan theorem
- Troelstra
- 1974
(Show Context)
Citation Context ...quantier-free choice for numbers is given by AC 0;0 -qf : 8x 0 9y 0 A 0 (x; y) ! 9f8xA 0 (x; fx); where A 0 is a quantier-free formula. 4 We also consider the binary Konig's lemma as formulated in [27]: WKL : 8f 1 (T (f) ^ 8x 0 9n 0 (lth(n) = 0 x ^ f(n) = 0 0) ! 9b 1 18x 0 (f(bx) = 0 0)); where b 1 1 : 8n(bn 1) and T (f) : 8n 0 ; m 0 (f(n m) = 0 ! f(n) = 0) ^ 8n 0 ; x 0 (f(n hxi) = 0 !... |

9 | On the arithmetical content of restricted forms of comprehension, choice and general uniform boundedness. Ann. Pure and Applied Logic 95
- Kohlenbach
- 1998
(Show Context)
Citation Context ...5.5 from [12]). Since furthermore PRA 2 E-G1A ! and { by [9] (section 4) { WKL can be derived in E-G1A ! +AC 1;0 -qf +F , 9 we obtain E-G1A ! + AC 1;0 -qf + F ` 0 1 -CA() ! A: Corollary 4.7 from [13] (combined with the elimination of extensionality procedure as used in the proof of corollary 4.5 in [13]) yields that G1A ! + 0 1 -IA ` A; and hence (since G1A ! + 0 1 -IA can easily be seen to be... |

8 | The use of a logical principle of uniform boundedness in analysis
- Kohlenbach
- 1995
(Show Context)
Citation Context ...e restricted complexity or rate of growth (compared to merely being primitive recursive). In fact in a series of papers we have shown that in many cases even a polynomial bound is guaranteed (see [9],=-=[11]-=-,[14] among others). 2) One can argue that PRA formalizes what has been calledsnitistic reasoning (see e.g. [26]). If the conservation of T over PRA has been establishedsnitistically (which is possibl... |

8 |
Proof-theoretic analysis of restricted induction schemata (abstract
- Parsons
- 1971
(Show Context)
Citation Context ...lows from 1) using the result from [19] that PRA+ 0 2 -IR proves every 0 3 -theorem of PRA+ 0 2 -IA and the fact that PRA 2 + 0 2 -IR PRA + 0 2 -IR. 3) follows from 2) and the fact (see e.g. [18]) that the provably recursive functions of PRA+ 0 2 {IA are just the functions denable in T 1 (i.e. the ( (! ! ) )-recursive functions) which includes the Ackermann function. 2 Remark 4.3 The only ... |

4 |
Recursion-theoretic hierarchies, Perspectives in Mathematical Logic
- Hinman
- 1978
(Show Context)
Citation Context ...ls and dening equations for all primitive recursive functionals of type level 2 in the sense of Kleene [7] (i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. [6](II.1) or [21]). We also have a schema of quantier-free induction (w.r.t. to this extended language) and -abstraction for number variables, i.e. (y:t[y])x = t[x]; x; y tuples of the same length. So... |

3 |
Systems of second-order arithmetic with restricted induction (abstract
- Friedman
- 1976
(Show Context)
Citation Context ....2 from the introduction. Remark 5.7 Proposition 5.5 in particular yields that relatively to EA 2 the principle PCM: 8f PCM(f) implies CA ar . For RCA 0 instead of EA 2 this implication is stated in [=-=-=-4]. A proof (which is dierent from our proof) can be found in [23]. Proposition 4.2 and proposition 5.5 together yield (using the fact thatsnitely many instances of 0 1 - d AC can be coded into a sin... |

3 |
Ordinal recursion in partial systems of number theory (abstract
- Parsons
- 1966
(Show Context)
Citation Context ... real numbers and the basic arithmetical operations and relations on them in EA 2 . The results of this section a fortiori hold for PRA 2 instead of EA 2 . 6 Here -recursive is meant in the sense of [=-=16-=-], i.e. unnested. In contrast to this the notion of - recursiveness as used e.g. in [2],[21] corresponds to nested recursion. 5 The representation of IR presupposes a representation of Q: Let j be the... |

2 |
Theories of type related to mathematical practice
- Feferman
- 1977
(Show Context)
Citation Context ...ed language) and -abstraction for number variables, i.e. (y:t[y])x = t[x]; x; y tuples of the same length. So PRA 2 is the second-order fragment of the (restricted)snite type system d PA ! j n from [3=-=]-=-. It is clear that the resulting system PRA 2 is conservative over PRA. We often write 1 instead of 0(0). Another option is to impose a restriction on the type-2-functionals which are allowed. We incl... |

1 |
Limited omniscience and the Bolzano{Weierstra principle
- Mandelkern
- 1988
(Show Context)
Citation Context ... i ` PCM ar ! 0 1 -IA: The other implication 0 1 -IA ! (PCM ar ) cannot be proved intuitionistically since (PCM ar ) implies the non{constructive so{called `limited principle of omniscience' (see [1=-=5]-=- for a discussion on this). 2) Proposition 5.2 provides much more information than corollary 5.3. In particular one can compute (in EA 2 ) uniformly in g a decreasing sequence of positive rational num... |

1 |
Quanti and one-quanti systems. Zap. naucn. sem
- unknown authors
- 1971
(Show Context)
Citation Context ...CM +BW +A-A is 0 3 -(but not 0 4 -)conservative over PRA+ 0 1 -IA and hence 0 2 -conservative over PRA. This also holds for E-PRA ! instead of PRA 2 . Together with the well-known fact (due to [16],[18],[19]) that the provable recursive functions of PRA+ 0 1 -IA are just the primitive recursive functions we obtain Corollary 1.3 The provably recursive functions of PRA 2 +AC 0;0 -qf+WKL+PCM +BW ... |

1 |
Quantifier-free and one-quantifier systems. Zap. naucn. sem
- unknown authors
- 1971
(Show Context)
Citation Context ...C0,0-qf+WKL+PCM− +BW−+A-A− is Π0 3-(but not Π04-)conservative over PRA+Σ0 1-IA and hence Π02-conservative over PRA. This also holds for E-PRAω − instead of PRA2− . Together withthewell-knownfact(dueto=-=[16]-=-,[18],[19])thattheprovablerecursivefunctions of PRA+Σ0 1-IA are just the primitive recursive functions we obtain Corollary 1.3 The provably recursive functions of PRA 2 −+AC 0,0 -qf+WKL+PCM − +BW − +A... |